Figures

Histological section showing cellular geometry in the anulus fibrosus. The section was taken from porcine tissue that was formalin-fixed, paraffin-embedded, and stained with Masson’s tri-chrome. The cells (marked by arrows) had an elongated shape and were aligned with the local collagen fiber direction.

Schematic representation of the boundary value problem. The matrix was assumed to be an infinite medium with transversely isotropic material properties. The direction of material symmetry corresponded to the direction of fiber orientation in the tissue and was parallel to the z axis of the cylindrical coordinate system (r,θ,z) shown. The cell was assumed to be a spheroidal region of distinct material properties centered at the origin, with radii given by a and b. At infinity, the boundary conditions were uniform normal stress tractions Tr and Tz. The cell and matrix were assumed to be perfectly bonded at their interface.

Strain magnitudes in the vicinity of a cell. The contour maps show values of strain components (εzz,εrr,εθθ, and εrz) for three cases: (A) a spherical cell in an isotropic matrix, (B) a spherical cell in an anisotropic matrix, and (C) an elongated cell in an anisotropic matrix (Table 2). Numbers on the plots indicate the values within the cell region. Values are presented for a far field strain of εzz0=0.05.

Effect of matrix anisotropy and cell geometry on strains within the cell. Values are presented for a far field strain of εzz0=0.05. The material properties used in these studies are summarized in Table 2. Arrows indicate the case of baseline isotropy. (A) The effect of anisotropy in the Young’s moduli, measured as the ratio Ezm/Erm. Results are shown for a spherical cell. (B) The effect of anisotropy in the shear moduli, measured as the ratio of Gzrm/Grθm. Results are shown for a spherical cell. (C) The effect of cell geometry, measured as the ratio a/b. The baseline anisotropic material properties were used in this study.

Volume changes in the vicinity of the cell. The contour maps show values of dilatation (e) for the same cases and boundary conditions as Fig. 3. Numbers on the plots indicate the values within the cell region.

Apparent membrane stretch. The surface areal strain within the plane of the cell–matrix interface is shown as a function of angular position along the cell membrane. Values are presented for the same cases and boundary conditions as Fig. 3.

Results for representative cases from the different intervertebral disc regions. The contour maps show values of the mean isotropic stress (σ̄) and the dilatation (e) for cases representative of the cell and matrix properties in the IVD (Tables 1 and 2). Values are presented for a far field strain of εzz0=0.05. Numbers on the plots indicate the values within the cell region. Values for σ̄ in the nucleus pulposus were scaled by a factor of 105, i.e., values in this region were actually on the order of 10 Pa.

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